Equivalent definitions of continuity of functions

This may seem trivial but, the following definitions of continuity of a function $f(x)$ on a metric space are equivalent -

1. $\forall \epsilon > 0$, $\exists \delta > 0$ such that for every point $x$,  $d(x,y) < \delta \implies d(f(x), f(y)) < \epsilon $
2. $x_n \rightarrow x  \implies f(x_n) \rightarrow f(x)$

I will just give the proof for $2) \implies 1)$, since the other way round is easier.

To find the $\delta$ for any $\epsilon > 0$ in $1)$, Consider the set $\mathcal{A}$ of all sequences $x_n \rightarrow x$ . Now, from the definition of continuity in $2)$ we know that for all sequences $f(x^i_n)  \rightarrow f(x)$ for $\{x^i_n\} \in \mathcal{A}$. Hence, $\exists N^i$ such that $\forall n>N^i$ , $d(f(x^i_n), f(x)) < \epsilon$. Let 

$ \delta = \arg_{x^i_N} \min_{x^i_n \in \mathcal{A}} d(x,x^i_N) $

Thus this value of $\delta$ can be used to find the neighbour-hood in $2)$.

Windows equivalents for Linux

One of the major factors that prevented me from using ubuntu regularly was the absence of a software as versatile as IDM for speeding up content download from the internet. Recently I found a great equivalent to IDM in ubuntu : axel

Here's how to use the axel command :

axel -N -n <number of connection> "<url>"

where the -N option is for not using the proxy server.

To download videos in linux its easy to use vlc and record the stream. A great pdf editor for linux is described here.